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Asymptotic Stability of Pseudo-simple Heteroclinic
Cycles in R
4Olga Podvigina, Pascal Chossat
To cite this version:
Olga Podvigina, Pascal Chossat. Asymptotic Stability of Pseudo-simple Heteroclinic Cycles in R4.
Asymptotic stability
of pseudo-simple heteroclinic cycles in
R
4
Olga Podvigina
Institute of Earthquake Prediction Theory
and Mathematical Geophysics
84/32 Profsoyuznaya St, 117997 Moscow, Russian Federation;
and
Pascal Chossat
Laboratoire J-A Dieudonn´
e, CNRS - UNS
Parc Valrose
06108 Nice Cedex 2, France
September 15, 2015
Abstract
Robust heteroclinic cycles in equivariant dynamical systems in R4 have been a sub-ject of intense scientific investigation because, unlike heteroclinic cycles inR3, they can have an intricate geometric structure and complex asymptotic stability properties that are not yet completely understood. In a recent work, we have compiled an exhaustive list of finite subgroups of O(4) admitting the so-called simple heteroclinic cycles, and have identified a new class which we have called pseudo-simple heteroclinic cycles. By contrast with simple heteroclinic cycles, a pseudo-simple one has at least one equilib-rium with an unstable manifold which has dimension 2 due to a symmetry. Here, we analyze the dynamics of nearby trajectories and asymptotic stability of pseudo-simple heteroclinic cycles in R4.
1
Introduction
It is known since the 80’s that vector fields defined on a vector space V (or more generally a Riemannian manifold), which commute with the action of a group Γ of isometries of V , can possess invariant sets which are structurally stable (within their Γ symmetry class) and which are composed of a sequence of saddle equilibria ξ1, . . . , ξM and a sequence of trajectories κi,
such that κi belong to the unstable manifold of ξi as well as to the stable manifold of ξi+1
and under certain conditions they can be asymptotically attracting. Robust heteroclinic cycles have been considerably studied in low-dimensional vector spaces [7, 2]. Although their properties and classification are well established in V = R3, the case of R4 appears to be much richer and yet not completely investigated. In [8] the so-called simple robust heteroclinic cycles were introduced, which can be defined as follows. Let ∆j denote the
isotropy subgroup of an equilibrium ξj. The heteroclinic cycle is simple if (i) the fixed point
subspace of ∆j, denoted F ix(∆j), is an axis; (ii) for each j, ξj is a saddle and ξj+1 is a sink in
an invariant plane F ix(Σj) (hence Σj ⊂ ∆j∩ ∆j+1); (iii) the isotypic decomposition1 for the
action of ∆j in the tangent space at ξj only contains one dimensional components. In [13]
we have found all finite groups Γ⊂ O(4) admitting simple heteroclinic cycles (i.e. for which there exists an open set of Γ-equivariant vector fields possessing such an invariant set). We have also pointed out that the definition of a simple heteroclinic cycle in former works [8, 9] had omitted condition (iii), which was implicitly assumed. If this condition is not satisfied, then the behaviour of the heteroclinic cycle turns out to be more complex, because there exists at least one equilibrium ξi at which the unstable manifold is two dimensional and,
moreover, is invariant under a faithful action of the dihedral group Dk for some k ≥ 3. We
called heteroclinic cycles satisfying (i) and (ii) but not (iii) pseudo-simple.
The aim of the present work is to investigate the dynamical properties of pseudo-simple heteroclinic cycles in R4. We shall therefore consider equations of the form
˙x = f (x), where f (γx) = γf (x) for all γ ∈ Γ, Γ ⊂ O(4) is finite (1) and f is a smooth map inR4, and which possess a pseudo-simple cycle.
Our main result stated in Theorem 1, section 3, is that if Γ ⊂ SO(4), then the pseudo-simple cycle is completely unstable. Namely, there exists a neighborhood of the heteroclinic cycle such that any solution with initial condition in this neighborhood, except in a subset of zero measure, leaves it in a finite time [11]. Then we aim at analyzing more deeply what the asymptotic dynamics can be in a neighbourhood of the heteroclinic cycle. We do this by focusing on specific examples. In Section 4 we introduce a subgroup of SO(4) which is algebraically elementary, however possessing enough structure to allow for the existence of pseudo-simple heteroclinic cycles. This group is a (reducible) representation of the dihedral group D3. Hence, Γ is algebraically isomorphic to D3. In this example, the cycle involves
two D3 invariant equilibria, ξ1 and ξ2, such that for both the linearization df (ξi) possess
a double eigenvalue with a D3 invariant associated eigenspace, one being negative and the
other being positive (unstable). We show that when the unstable double eigenvalue is small, an attracting periodic orbit can exist in the vicinity of the Γ-orbit of the heteroclinic cycle, the distance between the two invariant sets vanishes as the unstable eigenvalue tends to zero. This result is numerically illustrated by building an explicit third order system for which a pseudo-simple cycle exists. In the next section 5 we extend the group Γ by adding a generator which is a reflection in R4. The resulting group is isomorphic to D
3× Z2 and
possesses the same properties for the existence of a pseudo-simple cycle. We show that in
1The isotypic decomposition of the representation of a group is the (unique) decomposition in a sum of
this case the cycle is not completely unstable anymore, but asymptotically fragmentarily
stable [11] instead. This means that the cycle is not asymptotically stable, however in its
any small neighbourhood there exists a set of positive measure, such that any solution with initial condition in that set remains in this neighbourhood for all t > 0 and converges to the cycle as t→ ∞. This result is then illustrated numerically.
In Section 6 we present another example of a group in SO(4) admitting pseudo-simple cycles with a more complex group structure and acting irreducibly onR4. The theorem of existence
of nearby stable periodic orbits proven in Sec. 4 holds true for the same reasons as in the D3 case. We then investigate numerically the asymptotic behavior of trajectories near the
cycle and show that periodic orbits of various kinds can be observed as well as more complex dynamics when the unstable double eigenvalue becomes larger.
In the last section of this paper we discuss our findings and possible continuation of the study.
2
Pseudo-simple heteroclinic cycles
In this section we introduce the notion of a pseudo-simple heteroclinic cycle and notations which will be used throughout the paper.
2.1
Basic definitions
Given a finite group Γ of isometries of R4, let Σ be an isotropy subgroup of Γ (the subgroup of elements in Γ which fix a point in R4). We denote by F ix(Σ) the subspace comprised of
all points in R4, which are fixed by Σ.
Note that (i) if Σ ⊂ ∆ ⊂ Γ, then F ix(∆) ⊂ F ix(Σ); (ii) γ · F ix(Σ) = F ix(γΣγ−1) for any
γ ∈ Γ.
In the following we shall assume the Γ-equivariant system (1) admits a sequence of isotropy subgroups ∆j ⊂ Γ, j = 1, . . . , M, such that the following holds:
(i) dimF ix(∆j) = 1 (axis of symmetry). We denote Lj = F ix(∆j).
(ii) On each Lj, there exists an equilibrium ξj of (1).
(ii) For each j, Lj = Pj−1∩ Pj, where Pj is a plane of symmetry: Pj = F ix(Σj) for some
Σj ⊂ ∆j. We set PM = P0.
(iv) ξj is a sink in Pj−1 and a saddle in Pj. Moreover, a saddle-sink heteroclinic trajectory
κj of (1) connects ξj to ξj+1 in Pj (j = 1, . . . , M ). Note that κM connects ξM to ξ1 in
PM.
Conditions (i) - (iv) insure the existence of a robust heteroclinic cycle for (1). We denote by Jj the Jacobian matrix df (ξj). Since Lj is flow-invariant, Jj has a radial eigenvalue rj
with eigenvector along Lj and by (iv) we have rj < 0. Moreover (iv) implies that in Pj−1, Jj
corresponding to the unstable eigendirection in that plane. The remaining eigenvalue tj is
called transverse, its associated eigenspace is the complement to Pj−1⊕ Pj inR4.
Definition 1 We say that the group Γ admits robust heteroclinic cycles if there exists an
open subset of the set of smooth Γ-equivariant vector fields in Rn, such that vector fields in
this subset possess a (robust) heteroclinic cycle.
In order to insure the existence of a robust heteroclinic cycle it is enough to find m≤ M and γ ∈ Γ such that a minimal sequence of robust heteroclinic connections ξ1 → · · · → ξm+1
exists with ξm+1 = γξ1 (minimal in the sense that no equilibria in this sequence satisfy
ξi = γ′ξj for some γ′ ∈ Γ and 1 ≤ i, j ≤ m). It follows that γk = 1 where k is a divisor of M .
Definition 2 The sequence ξ1 → · · · → ξm and the element γ define a building block of the
heteroclinic cycle.
The asymptotic dynamics in a neighborhood of a robust heteroclinic cycle is what makes these objects special. A heteroclinic cycle is called asymptotically stable if it attracts all trajectories in its small neighborhood. When a heteroclinic cycle is not asymptotically stable, it can however possess residual types of stability. Below we define the notions, which are relevant in the context of this paper, see [11]. Given a heteroclinic cycle X and writing the flow of (1) as Φt(x), the δ-basin of attraction of X is the set
Bδ(X) ={x ∈ R4 ; d(Φt(x), X) < δ for all t > 0 and lim
t→+∞d(Φt(x), X) = 0}.
Definition 3 A heteroclinic cycle X is completely unstable if there exists δ > 0 such that
Bδ(X) has Lebesgue measure 0.
Definition 4 A heteroclinic cycle X is fragmentarily asymptotically stable if for any δ > 0,
Bδ(X) has positive Lebesgue measure.
2.2
Simple and pseudo-simple heteroclinic cycles
Recall that the isotypic decomposition of a representation T of a (finite) group G in a vector space V is the decomposition V = V(1)⊕ · · · ⊕ V(r) where r is the number of equivalence
classes of irreducible representations of G in V and each V(j) = T|Vj is the sum of the
equivalent irreducible representations in the j-th class. This decomposition is unique. The subspaces V(j) are mutually orthogonal (if G acts orthogonally).
Then the following holds [13].
Lemma 1 Let a robust heteroclinic cycle in R4 be such that for all j: (i) dim P
j = 2, (ii)
each connected component of Lj\ {0} is intersected at most at one point by the heteroclinic
cycle. Then the isotypic decomposition of the representation of ∆j in R4 is of one of the
following types:
2. Lj⊕⊥Vj ⊕⊥Wfj where fWj = Wj⊕ Tj has dimension 2.
3. Lj⊕⊥Vej ⊕⊥Wj where eVj = Vj ⊕ Tj has dimension 2.
In cases 2 and 3, ∆j acts in fWj (resp. eVj) as a dihedral group Dm in R2 for some m ≥ 3.
It follows that in case 2, ej is double (and ej = tj) while in case 3, −cj is double (and
−cj = tj).
Definition 5 A robust heteroclinic cycle in R4 satisfying the conditions 1 and 2 of lemma
1 is called simple if case 1 holds true for all j, and pseudo-simple otherwise.
Remark 1 Definition 5 implies that any pseudo-simple heteroclinic cycle in a Γ-equivariant
system (1), where Γ⊂ SO(4), has at least one 1 ≤ j ≤ M such that Σj ∼=Zk with k ≥ 3.
Remark 2 An order two element σ in SO(4) whose fixed point subspace is a plane P must
act as −Id in the plane P⊥ fully perpendicular to P . Nevertheless, to distinguish it from other rotations fixing the points on P , we call σ a plane reflection.
3
Instability of pseudo-simple cycles with Γ
⊂ SO(4)
In this section we prove the following theorem:
Theorem 1 Let ξ1 → . . . → ξM be a pseudo-simple heteroclinic cycle in a Γ-equivariant
system, where Γ⊂ SO(4). Then generically the cycle is completely unstable.
Proof: Following [8, 9, 11, 12], to study asymptotic stability of a heteroclinic cycle, we
approximate a ”first return map” on a transverse (Poincar´e) section of the cycle and consider its iterates for trajectories that stay in a small neighbourhood of the cycle.
In section 2.1 we have defined radial, contracting, expanding and transverse eigenvalues of the linearisation Jj = df (ξj). Let (˜u, ˜v, ˜w, ˜q) be coordinates in the coordinate system with
the origin at ξj and the basis comprised of the associated eigenvectors in the following order:
radial, contracting, expanding and transverse. If ˜δ is small, in a ˜δ-neighbourhood of ξj the
system (1) can be approximated by the linear system ˙u =−rju ˙v =−cjv ˙ w = ejw ˙ q = tjq. (2)
Here, (u, v, w, q) denote the scaled coordinates (u, v, w, q) = (˜u, ˜v, ˜w, ˜q)/˜δ. In fact a version
of the Hartman-Grobman theorem exists [2], which allows to linearize the system with a Ck
(k ≥ 1) equivariant change of variables in a neighbourhod of ξj if conditions of nonresonance
Let (u0, v0) be the point in Pj−1 where the trajectory κj−1 intersects with the circle
u2+ v2 = 1, and h be local coordinate in the line tangent to the circle at the point (u 0, v0).
We consider two crossections: e
Hj(in) ={(h, w, q) : |h|, |w|, |q| ≤ 1} and
e
Hj(out)={(u, v, w, q) : |u|, |v|, |q| ≤ 1, w = 1}.
Near ξj, trajectories of (1) with initial condition in eHj(in) hit the outgoing section eH
(out)
j
after a time τ , which tends to infinity as the initial condition comes closer to Pj−1. The
global map ˜ψj : eH
(out)
j → eH
(in)
j+1 associates a point where a trajectory crosses eH
(in)
j+1 with the
point where it previously crossed eHj(out). It is defined in a neighborhood of 0 (in the local coordinates in eHj(out)) and is a diffeomorphism. The first return map is now defined as the composition eg = gM ◦ . . . ◦ g1, where gj = ˜ψj◦ ˜ϕj.
As it is shown in [14, 11], in the study of stability only the coordinates that are in Pj⊥ are of importance. By ϕj and ψj we denote the maps ˜ϕj and ˜ψj restricted to Pj⊥(to be more
precise Hj(in)= Pj−1⊥ ∩ eHj(in) and Hj(out) = Pj⊥∩ eHj(out)).
Now, by definition a pseudo-simple cycle has at least one connection κj : ξj → ξj+1 such
that Σj ∼=Zk with k ≥ 3 (see remark 1). We can assume that j = 1. The equilibria ξ1 and
ξ2 belong to the plane P1 = Fix (Σ1). We now prove existence of ε > 0 such that
ϕ2◦ ψ1◦ ϕ1(H (in) 1 (ε))∩ H (out) 2 (ε) =∅, (3) where
H1(in)(ε) = {(w, q) ∈ H1(in) : |(w, q)| < ε} and H2(out)(ε) = {(v, q) ∈ H2(out) : |(v, q)| < ε}. Evidently, because of (3) the first return map in H1(in)(ε) cannot be completed in general, which implies that the cycle is completely unstable.
To prove (3) we employ polar coordinates (ρ1, θ1), resp. (ρ2, θ2), in H (out)
1 , resp. H (in) 2 , such
that v1 = ρ1cos θ1, q1 = ρ1sin θ1, w2 = ρ2cos θ2 and q2 = ρ2sin θ2. Due to (2) in the leading
order the maps ϕ1 and ϕ2 are
(ρ1, θ1) = ϕ1(w1, q1) = (v0,1w
c1/e1
1 , arctan(q1/v0,1)), (4)
(v2, q2) = ϕ2(ρ2, θ2) = (v0,2(ρ2cos θ2)c2/e2, tan θ2). (5)
In the leading order the global map ψ1 : H (out) 1 → H
(in)
2 is linear. The map commutes
with the group Σ2 acting on P2⊥ as rotation by 2π/k, where k ≥ 3, therefore in the leading
order ψ1 is a rotation by a finite angle composed with a linear transformation of ρ,
(ρ2, θ2) = ψ1(ρ1, θ1) = (Aρ1, θ1+ Θ), (6)
Denote α = min 1≥N≥2k|Θ − Nπ/k| (7) and set 0 < ε < min ( tanα 2, v0,1tan α 2 ) . (8) Any (w1, q1) ∈ H (in)
1 (ε) satisfies q1 < ε, therefore (4) and (8) imply that θ1 < α/2. Hence,
due to (6)-(8) |θ2 − Nπ/k| > α/2 for any k. The steady state ξ2 has k symmetric copies
(under the action of symmetries σ ∈ Σ2) of the heteroclinic connection κ2 : ξ2 → ξ3 which
belong to the hyperplanes θ2 = N π/k with some integer N ’s. Due to (5) and (8), the distance
of (v2, q2) to any of these hyperplanes is larger than tan(α/2), which implies (3). QED
4
Existence of nearby periodic orbits when Γ
⊂ SO(4):
an example
We have proved that any pseudo-simple heteroclinic cycle in a Γ-equivariant system, where Γ ⊂ SO(4), is completely unstable. In this section we show that nevertheless trajectories staying in a small neighbourhood of a pseudo-simple cycle for all t > 0 can exist. Namely, we give an example of a subgroup of SO(4) admitting pseudo-simple heteroclinic cycles and prove generic existence of an asymptotically stable periodic orbit near a heteroclinic cycle if an unstable double eigenvalue is sufficiently small.
4.1
Definition of Γ and existence of a pseudo-simple cycle
We write (x1, y1, x2, y2) ∈ R4 and zj = xj + iyj. Let Γ be the group generated by the
transformations
ρ : (z1, z2)7→ (z1, e
2πi 3 z
2), κ(z1, z2)7→ (¯z1, ¯z2) (9)
This group action obviously decomposes into the direct sum of three irreducible representa-tions of the dihedral group D3:
(i) the trivial representation acting on the component x1,
(ii) the one-dimensional representation acting on y1 by κy1 =−y1,
(iii) the two-dimensional natural representation ofD3 acting on z2 = (x2, y2).
There are two types of fixed-point subspaces and one type of invariant axis for this action: (i) P1 ={(x1, y1, 0, 0)} = F ix(ρ),
(ii) P2 ={(x1, 0, x2, 0)} = F ix(κ),
(iii) L = P1∩ P2 = F ix(D3).
Note that P1 is Γ-invariant, while P2 has two symmetric copies, P2′ = ρP2 and P2′′= ρ2P2. Proposition 1 The group Γ, generated by ρ and κ (9), admits pseudo-simple cycles (see
Proof: The proof of existence of an open set of smooth G equivariant vector fields with
saddle-sink orbits in P1 and in P2connecting equilibria ξ1 and ξ2 lying on the axis L and such
that ξ1 is a sink in P2 and ξ2 is a sink in P1, goes along the same arguments as in Lemma
5 of [13]. We just need to check that the cycle is pseudo-simple. This comes from the fact that the Jacobian matrix of the vector field taken at ξ1 or ξ2 has a double eigenvalue with
eigenspace {(0, 0, x2, y2)} due to the action of ρ, which generates the subgroup Z3 (negative
eigenvalue at ξ1 and positive eigenvalue at ξ2). QED
By letting Γ act on the heteroclinic cycle, one obtains a 6-element orbit of cycles which all have one heteroclinic connection in P1 and the other one in P2, P2′ or P2′′.
4.2
Existence and stability of periodic orbits
We prove the following theorem.
Theorem 2 Consider the Γ-equivariant system
˙x = f (x, µ), where f (γx, µ) = γf (x, µ) for all γ ∈ Γ, (10)
f : R4× R → R4 is a smooth map and Γ is generated by ρ and κ (9). Let ξ1 and ξ2 be the
equilibria introduced in proposition 1, −cj and ej be the eigenvalues of df (ξj), such that−c1
and e2 are double eigenvalues with a natural action of D3 in their eigenspaces and e1 and
−c2 are the eigenvalues associated with the one-dimensional eigenspace, where the action of
Γ in not trivial. Suppose that there exists µ0 > 0 such that
(i) e2 < 0 for −µ0 < µ < 0 and e2 > 0 for 0 < µ < µ0;
(ii) for any 0 < µ < µ0 there exist heteroclinic connections κ1 = (Wu(ξ1)∩P1)∩Ws(ξ2)̸= ∅
and κ2 = (Wu(ξ2)∩ P2)∩ Ws(ξ1)̸= ∅, introduced in proposition 1.
Denote by X the group orbit of heteroclinic connections κ1 and κ2:
X = (∪γ∈Γγκ1)
∪
(∪γ∈Γγκ2).
Then
(a) if 3c1 < e1 then there exist µ′ > 0 and δ > 0, such that for any 0 < µ < µ′ almost all
trajectories escape from Bδ(X) as t→ ∞;
(b) if 3c1 > e1 then generically there exists periodic orbit bifurcating from X at µ = 0. To be
more precise, for any δ > 0 we can find µ(δ) > 0 such that for all 0 < µ < µ(δ) the system (10) possesses an asymptotically stable periodic orbit that belongs to Bδ(X).
The rest of this section is devoted to the proof of the theorem. We start with two lemmas which describe some properties of trajectories of a generic D3-equivariant systems.
4.2.1 Two lemmas
A generic D3-equivariant second order dynamical system inC is
˙z = αz + β ¯z2. (11)
In polar coordinates, z = reiθ, it takes the form
˙r = αr + βr2cos 3θ,
˙
θ = −βr sin 3θ. (12)
We assume that α > 0 and β > 0. The system has three invariant axes with θ = Kπ/3,
K = 0, 1, 2, and three equilibria off origin, with r = α/β and θ = (2k + 1)π/3, k = 0, 1, 2.
We consider the system in the sector 0 ≤ θ < π/3, the complement part of C is related to this sector by symmetries of the group D3.
Trajectories of the system satisfy dr dθ =−
α + βr cos 3θ
β sin 3θ . (13)
Re-writing this equation as
dr dθ + r cos 3θ sin 3θ =− α β sin 3θ,
multiplying it by sin1/33θ and integrating, we obtain that
r sin1/33θ =−α
βS(θ) + C, where S(θ) =
∫ θ
0
sin−2/33θdθ, (14)
which implies that
r sin1/33θ + α βS(θ) = r0sin 1/33θ 0+ α βS(θ0) (15)
for the trajectory through the point (r0, θ0). Let θ(r; r0, θ0) be a solution to (12) with the
initial condition r(0) = r0 and θ(0) = θ0. Then
C = α βS(˜θ), where ˜θ = limr→0θ(r; r0, θ0). We can re-write (14) as r sin1/33θ =−α βS(θ) + α βS(˜θ). (16) Note, that S(π 3) = √ π 3 Γ(1/6)
Γ(2/3), where Γ is the Gamma function. (17)
Lemma 2 Given β > 0 and ε > 0, there exists α0 > 0 such that for any 0 < α < α0,
0 < r0 < ε and 0 < θ0 < π/3
ε
0
Figure 1: Trajectories of the system (12) in the sector 0≤ θ ≤ π/3.
Proof: Set α0 = εβ/4. From (12) d
dtr sin θ = ˙r sin θ + ˙θr cos θ = r sin θ(α− βr cos θ/2). (19) From (14), for 0 < θ0 < π/3 we have limr→∞θ(r; r0, θ0) = 0, which implies that d/dt(r sin θ) <
0 for large r. Therefore, the maximum of r sin θ is achieved either at t = 0, or at the point where 2α = βr cos θ. Since 0 < θ < π/3, at this point r sin θ = 2α sin θ/(β cos θ) < 4α/β < ε. Since r0 < ε, we have r sin θ < ε at t = 0 as well. (The behaviour of trajectories of the system
(12) is shown in figure 1.) QED
Lemma 3 Let τ (r0, θ0) denotes the time it takes the trajectory of the system (12) starting
at (r0, θ0) to reach r = 1 and ϑ(r0, θ0) denotes the value of θ at r = 1. Then
(i) τ (r0, 0) satisfies eατ (r0,0) = r0+ α/β r0(1 + α/β) . (ii) τ (r0, θ0) satisfies τ (r0, θ0) > τ (r0, 0) for any 0 < θ0 < π/3. (20) (iii) ϑ(r0, θ0) satisfies sin1/33ϑ(r0, θ0) + α βS(ϑ(r0, θ0)) = r0sin 1/33θ 0+ α βS(θ0). (21)
(iv) Given C > 0, β > 0 and 0 < θ0 < π/3, for sufficiently small α and r0
e−Cτ(r0,θ0) ≪ ϑ(r
(v) For small α, r0 ≪ α, r′ ≪ r0 and θ′ ≪ 1
|e−Cτ(r0+r′,θ0+θ′)− e−Cτ(r0,θ0)| < e−Cτ(r0,θ0)| r
′
αr0
+ 3θ′S(θ0)τ (r0, θ0)|.
Proof: To obtain (i), we integrate the first equation in (12), where we set θ = 0. Since in
(12) θ < θ′ implies ˙r(r, θ) > ˙r(r, θ′), (ii) holds true. The equality (iii) follows from (15). Since ˙θ < 0 (see (12) ), (iii) implies that ϑ(r0, θ0)≪ 1 for small α and r0. Therefore
ϑ(r0, θ0)≈ 1 3 ( r0sin1/33θ0+ α βS(θ0) )3 > 1 3β3 ( max(α, βr0) min(sin1/33θ0, S(θ0)) )3 . (22)
Due to (ii), to prove (iv) it is sufficient to show that e−Cτ(r0,0) ≪ ϑ(r
0, θ0).
Because of (i), for small r0 and α
e−Cτ(r0,0) = ( r0(1 + α/β) r0 + α/β )C/α ≈ [(1 − s)1/s]C/(α+βr0), where s = α α + βr0 .
Since sup0<s<1(1− s)1/s= e−1, we have
e−Cτ(r0,0) < e−C/ max(α,βr0). (23) Together, (22) and (23) imply (iv).
For r ≪ α the first equation in (12) can be approximated by ˙r = αr. Therefore, the difference ∆τ (r′) = τ (r0+ r′, θ0)− τ(r0, θ0) satisfies
r0eα∆τ (r
′)
≈ (r0 + r′),
which implies ∆τ (r′)≈ r′/αr0. Hence,
e−Cτ(r0+r′,θ0)− e−Cτ(r0,θ0) ≈ e−Cτ(r0,θ0) r
′
αr0
. (24)
Consider two nearby trajectories, θ(r; r0, θ0) and θ(r; r0, θ0 + θ′). From (12),
θ(r; r0, θ0+ θ′)− θ(r; r0, θ0) < θ′ for all t > 0, therefore in view of (iii)
| ˙r(r, θ(r; r0, θ0+ θ′))− ˙r(r, θ(r; r0, θ0))| | ˙r(r, θ(r; r0, θ0))| < |3θ ′βr2sin 3θ| |αr + βr2cos 3θ| <| 3β α θ ′r sin1/33θ| < |3θ′S(θ 0)|,
which implies that
|τ(r0, θ0+ θ′)− τ(r0, θ0)| < |3θ′S(θ0)τ (r0, θ0)|.
Hence,
|e−C(τ(r0,θ0+θ′))− e−Cτ(r0,θ0)| < |e−Cτ(r0,θ0)3θ′S(θ
0)τ (r0, θ0)|. (25)
4.2.2 Proof of Theorem 2
As in the proof of theorem 1, we approximate trajectories in the vicinity of the cycle by superposition of local and global maps. We consider g = ϕ1ψ2ϕ2ψ1 : H
(out) 1 → H
(out)
1 , where
the map ϕ1 is given by (4). The map ψ2 in the leading order is linear:
(w1, z1) = ψ2(v2, z2) = (B11v2+ B12z2, B21v2+ B22z2). (26)
Because of (i), for small µ the expanding eigenvalue of ξ2 depends linearly on µ, therefore
without restriction of generality we can assume that e2 = µ. Generically, all other eigenvalues
and coefficients in the expressions for local and global maps do not vanish for sufficiently small µ and are of the order of one. We assume them to be constants independent of µ. From (ii), the eigenvalues satisfy e1 > 0, −c1 < 0 and−c2 < 0.
For small enough ˜δ, in the scaled neighbourhoods Bδ˜(ξ2) the restriction of the system
to the unstable manifold of ξ2 in the leading order is ˙s = µs + β ¯s2, where we have denoted
s = w2 + iq2. As in the proof of theorem 1, (ρ1, θ1) and (ρ2, θ2) denote polar coordinates
in H1(out) and H2(in), respectively, such that v1 = ρ1cos θ1, q1 = ρ1sin θ1, w2 = ρ2cos θ2 and
q2 = ρ2sin θ2. We assume that the local bases near ξ1and ξ2are chosen in such a way that the
heteroclinic connection ξ2 → ξ1 goes along the directions arg(θj) = 0 for both j = 1, 2, which
implies β > 0. In the complement subspace the system is approximated by the contractions ˙u = −r2u and ˙v = −c2v. In terms of the functions τ (r, θ) and ϑ(r, θ) introduced in lemma
3, the map ϕ2 is
(v2, q2) = ϕ2(ρ2, θ2) = (v0,2e−c2τ (ρ2,θ2), sin ϑ(ρ2, θ2)).
In physical space, there are two heteroclinic trajectories from ξ1 to ξ2, with positive or
negative w1, and three trajectories from ξ2 to ξ1, with θ2 = 0, 2π/3 or 4π/3. Let H (out) 1 be
the crossection of the heteroclinic trajectory ξ1 → ξ2 with positive w1. We take a modified,
compared to (6), map ψ1:
(ρ2, θ2) = ψ1(ρ1, θ1) = γ(Aρ1, θ1+ Θ), where γ = κlρs. (27)
By choosing s, we obtain
(ρ2, θ2) = ψ1(ρ1, θ1) = (Aρ1, (−1)lΘ′), where Θ′ = Θ + 2πs/3 satisfies − π/3 < Θ′ < π/3.
(28) According to lemma 3(iv), for small ρ2 and µ
e−c2τ (ρ2,θ2)≪ sin ϑ(ρ
2, θ2),
therefore we have ψ2(v2, q2) ≈ (B12q2, B22q2). For small θ1 we have sin θ1 ≈ tan θ1 ≈ θ1.
Taking into account (4), (28), lemma 2 and lemma 3(iii), we obtain that
g(ρ1, θ1)≈
(
C1(ρ1Aβ sin1/33 ˜Θ + µS( ˜Θ))3c1/e1, C2(ρ1Aβ sin1/33 ˜Θ + µS( ˜Θ))
)
where C1 = v0,13−1β−3|B12|c1/e1 and C2 = (−1)lv−10,13−1β−3B22. (Here, we have denoted
˜
Θ =|Θ′| and the power l in (28) is chosen in such a way that in (w1, q1) = ψ2ϕ2ψ1(ρ1, 0) the
value of w1 is positive for small ρ1.)
(a) From (29), the ρ-component of g satisfies
gρ(ρ1, θ1) > C3ρ 3c1/e1
1 , where C3 = C1(Aβ sin1/3Θ)˜ 3c1/e1,
hence if 3c1 < e1 then for any 0 < δ < C
e1/(e1−3c1)
3 the iterates gn(ρ1, θ1) with initial
0 < ρ1 < δ satisfy gnρ(ρ1, θ1) > δ for sufficiently large n.
(b) Assume that 3c1 > e1. Since ρ1 = 0 is a solution to
gρ(ρ1, θ1)− ρ1 = 0 (30)
for µ = 0, by the implicit function theorem, the equation (30) has a unique solution ρ1(µ),
0 ≤ µ ≤ µ1, for some µ1 > 0 and dr1/dµ = 3c1C1µ(3c1−e1)/e1S3c1/e1( ˜Θ)/e1 at µ = 0. (In
order to apply the implicit function theorem, we define the function h(ρ1, µ) = ρ1−gρ(ρ1, θ1)
for negative ρ1 by setting h(−ρ1, µ) = −h(ρ1, µ). Note, that gρ(ρ1, θ1) does not depend
on θ1.) Therefore, for small µ the fixed point of the map g (29) can be approximated by
(ρp, θp) = (C1(µS( ˜Θ))3c1/e1, C2µS( ˜Θ)). This fixed point is an intersection of a periodic orbit
with H1(out). The distance from (ρp, θp) to X depends on µ as µ3c1/e1, therefore the trajectory
approaches X as µ→ 0. For a given small δ > 0, to find µ(δ) we approximate trajectories near κ1 and κ2 by linear (in (v1, q1) and (v2, q2), respectively) maps, near ξ1 we use the
approximation (2) and near ξ2 we employ lemma 2. We do not go into details.
To study stability of the fixed point (ρp, θp), we consider the difference g(ρ1+ r′, θ1+ θ′)−
g(ρ1, θ1), assuming that r′ and θ′ are small and (ρ1, θ1) are close to (ρp, θp). For the maps
ψ1, ψ2 and ϕ1 we have ψ1(ρ1+ r′, θ1+ θ′)− ψ1(91, θ1) = (Ar′, θ′), ψ2(v2+ v′, q2+ q′)− ψ2(v2, q2) = (B11v′ + B12q′, B21v′ + B22q′), ϕ1(w1+ w′, q1+ q′)− ϕ1(w1, q1) = ((v0,1c1/e1)w c1/e1−1 1 w′, (1 + (q1/v0,1)2)−1v0,1−1q′). (31)
Recall that (see lemma 3(v))
|e−c2(τ (ρ2+r′,θ2+θ′))− e−c2τ (ρ2,θ2)| < e−c2τ (ρ2,θ2)| r
′
µρ2
+ 3θ′S(θ2)τ (ρ2, θ2)|. (32)
From (21), for small µ and ρ2 ≪ µ
|ϑ(ρ2+ r′, θ2+ θ′)− ϑ(ρ2, θ2)| ≈ ϑ2/3(ρ2, θ2)(r′sin1/33 ˜Θ + θ′µ sin−2/33 ˜Θ/β). (33)
By arguments similar to the ones applied in the proof of lemma 3(iv), for small µ and ρ2 the
r.h.s. of (32) is asymptotically smaller than the r.h.s. of (33). Therefore, combining (33) with (31), we obtain that for small µ and ρ1 ≪ µ
where the constants Aj depend on v0,1, e1, c1, β, B12, B22 and ˜Θ. Since 3e1 > c1, for small
µ we have
|g(ρ1+ r′, θ1+ θ′)− g(ρ1, θ1)| < µq|(r′, θ′)|,
and q = min((3c1− e1)/(2e1), 1), which implies that the bifurcating periodic orbit is
asymp-totically stable. QED
Remark 3 In the proof of the theorem we have considered the map g : H1(out) → H1(out), g = ϕ1ψ2ϕ2ψ1, where ψ1 involves the symmetry γ = κlρsand the choice of l and s depends on
Θ and B12. Hence, depending on the values of these constants, there can exist geometrically
different periodic orbits in the vicinity of the group orbit of the heteroclinic cycle ξ1 → ξ2 → ξ1
with different number of symmetric copies of heteroclinic connections κ1 and κ2. The cycle
shown in figure 2 involves two symmetric copies of the connection κ1 and one connection
κ2. For a different group Γ ⊂ SO(4) considered in section 6, we present examples of various
periodic orbits which are obtained by varying the coefficients of the respective normal form, see figure 4.
4.3
A numerical example
In this subsection we present an example of a Γ-equivariant vector field possessing a hetero-clinic cycle, introduced in proposition 1, where the expanding eigenvalue of df (ξ2) is small.
In agreement with theorem 2, asymptotically stable periodic orbits exist in the vicinity of the cycle. To construct the numerical example, we start from a lemma that determines the structure of Γ-equivariant vector fields (the proof is left to the reader):
Lemma 4 Any Γ-equivariant vector field is
˙z1 = An1n2n3n4z n1 1 z¯ n2 1 z n3 2 z¯ n4 2 ˙z2 = Bn1n2n3n4z n1 1 z¯ n2 1 z n3 2 z¯ n4 2 , (34)
where An1n2n3n4 and Bn1n2n3n4 are real, An1n2n3n4 ̸= 0 whenever n3− n4 = 0( mod 3) and
Bn1n2n3n4 ̸= 0 whenever n3 − n4 = 1( mod 3).
Keeping in (34) all terms of order one and two, and several terms of the third order, we consider the following vector field:
˙z1 = a1z1+ a2z¯1+ a3z12+ a4z¯12+ a5z1z¯1 + a6z2z¯2+ a7z12z¯1+ a8z1z2z¯2+ a9z23+ a10z¯23
˙z2 = b1z2+ b2z1z2+ b3z¯1z2 + b4z¯22+ b5z2z1z¯1+ b6z22z¯2,
(35) where
we set a3+ a4+ a5 = 0, a7 = a8 =−1, and denote α = a1+ a2, α′ = a1− a2. (36)
The system (34) restricted into L = FixD3 is
which implies that the steady states ξ1 with x1 = α1/2 and ξ2 = −ξ1 exist whenever α > 0.
The radial eigenvalues of ξj are −2α. The non-radial eigenvalues of df(ξ1) are
−c1 = a1 − a2+ 2(a3− a4)
√
α− α along the eigendirection y1
e1 = b1+ (b2+ b3)
√
α + b5α along the eigendirections x2 and y2
(37) and the eigenvalues at df (ξ2) are
e2 = a1− a2− 2(a3− a4)
√
α− α along the eigendirection y1
−c2 = b1− (b2+ b3)
√
α + b5α along the eigendirections x2 and y2
(38) The eigenvalues e1 and −c2 are double.
With (36) taken into account, the restriction of the system (35) into the plane P1 is
˙x1 = αx1− 2(a3+ a4)y12− x1(x21+ y12)− x1x22
˙
y1 = α′y1 + 2(a3− a4)x1y1− y1(x21+ y21)
(39) and the one into the plane P2 is
˙x1 = αx1+ a6x22− x31− x1x22+ (a9+ a10)x32
˙x2 = b1x2+ (b2+ b3)x1x2+ b4x22+ b5x21x2+ b6x32
(40) In both planes the system has a form which is well-known to produce saddle-sink connections between equilibria ξ1 and ξ2 if certain conditions are fulfilled, see e.g. [1]. A sufficient
condition for existence of a heteroclinic connection ξ1 → ξ2 in P1 is
a3+ a4 > 0, a3− a4 > 0, a3− a4+ ((a3− a4)2+ α′)1/2 < α1/2. (41)
The expression for a condition of the existence of a connection ξ2 → ξ1 in P2 is too bulky,
and therefore is not presented.
We choose the values of the coefficients so that the conditions for existence of heteroclinic connections in P1 and P2 are satisfied, with ξ1 being a saddle in P1 and a sink in P2 and vice
versa for ξ2. We also choose the coefficients such that the stability condition 3c1 > e1 (see
theorem 2) is satisfied.
Finally we adjust coefficients so that the unstable, double eigenvalue e2 at ξ2 verifies is small.
The simulations are performed with the following values of coefficients:
α = 0.3, α′ = 0.2, a3 = 0.3, a4 =−0.05, a5 =−0.25, a6 = 0.6,
a7 = a8 =−1, a9 = 0.1, a10 = 0.15;
b1 = 0.2, b2 =−0.1, b3 =−0.09, b4 =−0.1, b5 =−1, b6 =−1,
(42)
which implies that the eigenvalues are −c1 = −0.2041, e1 = 0.2834, −c2 = −0.4834, e2 =
0.0041.
On figure 2 we show projection of the periodic orbit and of the group orbit of heteroclinic connections, comprising the cycle, on a plane in R4, and time series of x
1 and y1. The
periodic orbit follows two heteroclinic connections ξ1 → ξ2 in P1, and only one ξ2 → ξ1 in
P2, see remark 3.
In Section 6 we shall explore numerically the dynamics near the pseudo-simple hetero-clinic cycle with a more involved symmetry subgroup of SO(4).
(a)
10000
12000
14000
16000
18000
20000
t
(b)
x
1y
1Figure 2: Projection of the periodic orbits (solid line) and of the heteroclinic connec-tions ξ2 → ξ1 (dashed lines) into the plane < v1, v2 >, where v1 = (4, 2, 4, 1.5) and v2 = (2, 4,−1.5, 4), (a). The steady state ξ1 is denoted by a hollow circle and ξ2 by filled
5
Stability of pseudo-simple heteroclinic cycles when
Γ
⊂ O(4), but Γ ̸⊂ SO(4)
Here we show that presence of a reflection in the group Γ can completely change the asymp-totic dynamics of the system (1) near a pseudo-simple heteroclinic cycle. As in the previous section we hold on an example, which is built as an extension of the group Γ generated by transformations (9). Let’s introduce the reflection
σ : (z1, z2)7→ (z1, ¯z2) (43)
and define ˜Γ = Γ∪ σΓ. This group admits the same one-dimensional and two-dimensional fixed point subspaces as Γ, therefore it also admits (similar) pseudo-simple heteroclinic cycles. However, it also has two types of three-dimensional fixed point subspaces:
(i) V ={(x1, y1, x2, 0)} = F ix(σ),
(ii) W ={(x1, 0, x2, y2)} = F ix(σκ).
Note that (i) V = P1⊕ P2 (the invariant planes introduces in Section 4.1) and (ii) V has
two symmetric copies V′ = ρV and V′′= ρ2V , while W is a singleton. It follows from point
(i) that if it exists, the heteroclinic cycle lies entirely in V .
5.1
A theorem
Theorem 3 Consider the ˜Γ-equivariant system
˙x = f (x), where f (γx) = γf (x) for all γ∈ ˜Γ (44)
and f is a smooth map. Suppose that the system possesses equilibria ξ1 and ξ2 and a
pseudo-simple heteroclinic cycle, introduced in proposition 1, where the heteroclinic connection ξ2 →
ξ1 belongs to the half-plane of P2 satisfying x2 > 0. We assume that ∂2fx2/∂x
2
2 > 0, where
fx2 denotes the x2-component of f . Let −cj and ej be the stable and unstable eigenvalues of
df (ξj), respectively, such that −c1 and e2 have multiplicity 2.
Then, for sufficiently small expanding eigenvalue e2 the cycle is f.a.s. (see definition 4).
We begin with a proof of a lemma about properties of trajectories in a generic D3
-equivariant second order dynamical system (11), where we assume that α > 0 and β > 0. Recall, that in polar coordinates the system takes the form (12), the system is considered in the sector 0≤ θ ≤ π/3, by τ(r0, θ0) we have denoted the time it takes the trajectory of the
system (12) starting at (r0, θ0) to reach r = 1 and by ϑ(r0, θ0) the value of θ at r = 1.
Lemma 5 If 0 < θ0 < π/6 then ϑ(r0, θ0) satisfies
sin 3ϑ(r0, θ0) < ( r0β cos 3θ0+ α β cos 3θ0 + α )3 sin 3θ0. (45)
Proof: From (12),
d dt(r
6sin23θ) = 6αr6sin23θ,
which implies that
r6(t) sin23θ(t) = e6αtr60sin23θ0.
Setting t = τ (r0, θ0) we obtain that
sin 3θ(τ ) = e3ατr30sin 3θ0. (46)
As well, (12) implies that ˙θ < 0 and that for θ1 < θ2 we have ˙r(r, θ1) > ˙r(r, θ2). Hence, the
trajectory (r(t), θ(t)) starting at r(0) = r0 and θ(0) = θ0 satisfies
˙r(r(t), θ(t)) > ˙r(r(t), θ0) = αr + β cos 3θ0r2 for all t > 0.
Therefore, by the same arguments as we employed in the proof of lemma 3(i,ii) eατ (r0,θ0) < r0β cos 3θ0+ α
r0(β cos 3θ0+ α)
. (47)
Combining (46) and (47) we obtain statement of the lemma. QED
Proof of the Theorem: We aim at finding conditions such that the first return map
g = ψ1ϕ1ψ2ϕ2 : H2(in) → H (in)
2 is a contraction and hence converges to 0 as the number
of iterates tends to infinity. (Note, that here we consider the map H2(in) → H2(in), while in theorems 1 and 2 it was H1(out) → H1(out).) The local and global maps are defined the same way, as it was in the proofs of theorems 1 and 2. Because of the reflection σ, the constants Θ, B12 and B21 in (6) and (26) vanish, hence global maps take the form
(ρ2, θ2) = ψ1(ρ1, θ1) = (Aρ1, θ1), (w1, q1) = ψ2(v2, q2) = (B11v2, B22q2). (48)
Recall, that (ρ1, θ1) and (ρ2, θ2) denote polar coordinates in H (out)
1 and H
(in)
2 , respectively,
such that v1 = ρ1cos θ1, q1 = ρ1sin θ1, w2 = ρ2cos θ2 and q2 = ρ2sin θ2. The map ϕ1 is given
by (4). The map ϕ2 is
(v2, q2) = ϕ2(ρ2, θ2) = (v0,2e−c2τ (ρ2,θ2), tan ϑ(ρ2, θ2)).
The composition g = ψ1ϕ1ψ2ϕ2 therefore is
g(ρ2, θ2) = ( Av0,1 ( B11v0,2e−c2τ (ρ2,θ2) )c1/e1 , arcsin(B22tan ϑ(ρ2, θ2) ) . (49) The condition ∂2f x2/∂x 2
2 > 0 implies that the restriction of the system (44) into the
unstable manifold of ξ2, and which is of the form (11), satisfies β > 0. The condition α > 0
estimate τ (ρ2, θ2) and ϑ(ρ2, θ2). Denote by let gρ and gθ the first and second components of
g in (49). From lemmas 3(i,ii) and 5 we can write the estimates gρ(ρ2, θ2)≤ C ( ρ2(β+e2) ρ2β+e2 )c1c2/e1e2 gθ(r2, θ2)≤ arcsin ( B22tan ( 1/3 arcsin [( ρ2β cos 3θ2+e2 β cos 3θ2+e2 )3 · sin(3θ0) ])) (50)
where C = Av0,1(B11v0,2)c1/e1. We show that gρand gθ are contractions under the conditions
stated in the theorem.
The function gρ(ρ2, θ2) is a smooth function of ρ2 for all ρ2 > 0. Its derivative is positive
and has the expression
gρ′(ρ2) = hC
e2(β + e2)h
(ρ2β + e2)h+1
ρh2−1,
where we have denoted h = c1c2
e1e2. The condition h > 1 implies that limρ→0g
′
ρ(ρ) = 0. Therefore,
there exists η > 0 such that gρ′(ρ2) < 1 and gρ is a contraction when ρ2 < η.
To prove that gθ is a contraction let us assume that θ2is chosen small enough to allow for the
approximation tan 3θ2 ≈ sin 3θ2 ≈ 3θ2 (this is not necessary but simplifies the calculations).
Then gθ(θ2)≤ B22 ( ρ2β cos 3θ2+ e2 β cos 3θ2+ e2 )3 θ2.
If we chose e2 and ρ2 small enough (β is a fixed positive constant), then the factor in front
of θ2 is smaller than 1. Therefore, if ρ2 converges to 0 by iterations of g, the same is true for
θ2. Combining the conditions for gρ and gθ we get the result. QED
Remark that the convergence to 0 of θ2 is slow compared to that of ρ2.
Remark 4 Suppose that, similarly to theorem 2, we consider a ˜Γ-equivariant system which
depends on a parameter µ, possesses a heteroclinic cycle for 0 < µ < µ0 and e2 = 0 for
µ = 0. Then by theorem 3 there exists µ1 > 0 such that the heteroclinic cycle is f.a.s. for
0 < µ < µ1. Moreover, the condition ∂2fx2/∂x
2
2 > 0 (which is equivalent to β > 0) is always
satisfied in such a system, by the reasons given in the proof of theorem 2. Note however, that closeness to the bifurcation point, and even smallness of e2, are not necessary for fragmentary
asymptotic stability of the cycle.
5.2
A numerical illustration of Theorem 3
The equivariant structure of ˜Γ-equivariant vector fields is easily deduced from that of Γ equivariant vector fields.
Lemma 6 Any ˜Γ-equivariant vector field has the form (34) (see Lemma 4) with the following
In this subsection we consider the same system (34) as we considered in subsection 4.3. Because of the presence of the symmetry σ (see lemma 6), coefficients of this system satisfy
a9 = a10 and b2 = b3. As in subsection 4.3, we denote α = a1+ a2, α′ = a1− a2 and assume
a3+ a4+ a5 = 0 and a7 = a8 =−1. Therefore, the system is
˙z1 = a1z1+ a2z¯1+ a3z12+ a4z¯12+ a5z1z¯1+ a6z2z¯2 − z21z¯1− z1z2z¯2+ a9(z23+ ¯z23)
˙z2 = b1z2+ b2(z1z2+ ¯z1z2) + b4z¯22+ b5z2z1z¯1+ b6z22z¯2,
(51) The conditions for the existence of the heteroclinic cycle in the invariant space V are similar to those given in Section 4.3. It is however interesting to analyze the dynamics in the invariant space W . Consider the restriction of system (51) into the subspace W :
˙x1 = αx1+ a3(x22+ y22)− x31− x1(x22+ y22) + 2a9(x32− 3x2y22)
˙x2 = b1x2+ 2b2x1x2 + b4(x22− y22) + b5x2x21+ b6x2(x22+ y22)
˙
y2 = b1y2 + 2b2x1y2− 2b4x2y2+ b5y2x21+ b6y2(x22+ y22).
(52)
In the limit b4 = 0 any plane containing the axis L is flow-invariant by (52). In fact, any
such plane is a copy of P2 by some rotation around L. Therefore, whenever a saddle-sink
connection exists in P2, it generates a two-dimensional manifold of saddle-sink connections
in W (an ”ellipsoid” of connections). Small ˜Γ-equivariant perturbations of the system, in particular switching on b4 ̸= 0, do not destroy this manifold. If in addition the conditions
(41) are satisfied, then a continuum of robust heteroclinic cycles connecting ξ1 and ξ2 exists
for the ˜Γ-equivariant vector field.
We take the same values of the coefficients as in (42), except that a9 = a10 = 0.1 and
b2 = b3 =−0.6 so that the system is ˜Γ-equivariant.
Note, that when the symmetry σ is absent, the heteroclinic cycle is completely unstable but a nearby asymptotically stable periodic orbit is observed exists, in accordance with the results of previous sections. When the symmetry σ is present, the trajectory starting from some point in a neighborhood of ξ2 asymptotically converges to the heteroclinic cycle (or to
its symmetric copy) as t→ ∞.
6
Another example with a group Γ
⊂ SO(4)
The groups which have been considered in sections 4 and 5 had the advantage of allow-ing for an elementary algebraic description, which gives an opportunity to study in detail the structure on invariant subspaces and related dynamics. In this section we aim at pro-viding an example of an admissible group with pseudo-simple heteroclinic cycles, with a non-elementary group structure. In principle, all such groups can be listed as we did for admissible groups with simple heteroclinic cycles in [13].
We then build an equivariant system for this group and study numerically its asymptotic behavior near the pseudo-simple heteroclinic cycle. We begin with a description of the bi-quaternionic presentation of finite subgroups of SO(4) together with some geometrical properties of these actions, which we employ to identify invariant subspaces.
(a)
x
1x
1(b)
(c)
y
1x
2Figure 3: Projection of a trajectory of the system (51) converging to a heteroclinic cycle (solid line) and of heteroclinic connections (dotted, dashed and dashed-dotted lines, different types of lines label different group orbits of heteroclinic connections) into the plane < v1, v2 >,
where v1 = (4,−2, 4, 1.5) and v2 = (2, 4,−1.5, 4), (a). Projection of this trajectory into the
6.1
Subgroups of SO(4): presentation and notation
Here we briefly introduce the bi-quaternionic presentation and notation for finite subgroups of
SO(4), which has been used in [13] to determine all admissible groups for simple heteroclinic
cycles in R4. See [3] for more details.
A real quaternion is a set of four real numbers, q = (q1, q2, q3, q4). Multiplication of
quaternions is defined by the relation
qw = (q1w1− q2w2− q3w3− q4w4, q1w2+ q2w1+ q3w4 − q4w3,
q1w3− q2w4+ q3w1− q4w2, q1w4+ q2w3− q3w2+ q4w1).
(53) ˜
q = (q1,−q2,−q3,−q4) is the conjugate of q, and q˜q = q12+ q22+ q23+ q24 =|q|2 is the square
of the norm of q. Hence ˜q is also the inverse q−1 of a unit quaternion q. We denote by Q the multiplicative group of unit quaternions; obviously, its unity element is (1, 0, 0, 0).
Due to existence of a 2-to-1 homomorphism of Q onto SO(3) (see [3]), finite subgroups of Q are named after respective subgroups of SO(3). They are:
Zn = ⊕nr=0−1(cos 2rπ/n, 0, 0, sin 2rπ/n) , Dn =Z2n⊕ ⊕2nr=0−1(0, cos rπ/n, sin rπ/n, 0)
V = ((±1, 0, 0, 0)) , T = V ⊕ (±1 2,± 1 2,± 1 2,± 1 2) O = T ⊕√1 2((±1, ±1, 0, 0)) , I = T ⊕ 1 2((±τ, ±1, ±τ−1, 0)) (54) where τ = (√5 + 1)/2. Double parenthesis denote all possible permutations of quantities within the parenthesis and for I only even permutations of (±τ, ±1, ±τ−1, 0) are elements of
the group.
By regarding the four numbers (q1, q2, q3, q4) as Euclidean coordinates of a point in R4,
any pair of unit quaternions (l, r) can be related to the transformation q→ lqr−1, which is an element of the group SO(4). The respective mapping Φ :Q×Q → SO(4) is a homomorphism onto, whose kernel consists of two elements, (1, 1) and (−1, −1). Therefore, a finite subgroup of SO(4) is a subgroup of a product of two finite subgroups of Q. There is however an additional subtlety. Let Γ be a finite subgroup of SO(4) and G = Φ−1(Γ) = {(lj, rj), 1 ≤
j ≤ J}. Denote by L and R the finite subgroups of Q generated by lj and rj, 1 ≤ j ≤ J,
respectively. To any element l ∈ L there are several corresponding elements ri, such that
(l, ri)∈ Q, and similarly for any r ∈ R. This establishes a correspondence between L and
R. The subgroups of L and R corresponding to the unit elements in R and L are denoted
by LK and RK, respectively, and the group G by (L | LK; R| RK).
Lemma 7 (see proof in [12]) Let N1 and N2 be two planes in R4 and pj, j = 1, 2, be the
elements of SO(4) which act on Nj as identity, and on Nj⊥ as−I, and Φ−1pj = (lj, rj), where
Φ is the homomorphism defined above. Denote by (l1l2)1 and (r1r2)1 the first components of
the respective quaternion products. The planes N1 and N2 intersect if and only if (l1l2)1 =
Lemma 8 Consider g ∈ SO(4), Φ−1g = ((cos α, sin αv); (cos β, sin βw)). Then dim Fix < g >= 2 if and only if cos α = cos β.
Lemma 9 Consider g, s∈ SO(4), where Φ−1g = ((cos α, sin αv); (cos α, sin αw)) and
Φ−1s = ((0, v); (0, w)).
Then Fix < g >= Fix < s >.
6.2
The group Γ = (
D
3| Z
2;
O | V)
The group (D3| Z2;O | V) was proposed in [13] as an example of a subgroup of O(4)
ad-mitting pseudo-simple heteroclinic cycles. It is the (unique) four dimensional irreducible representation of the group GL(2, 3) (2× 2 invertible matrices over the field Z3). This group
is generated by the elements σ1 (order 8) and σ2 (order 2) below:
σ1 = ( 0 2 2 2 ) , σ2 = ( 2 0 0 1 ) In quaternionic form its elements are:
((1, 0, 0, 0) ; V), ((cos(π/3), 0, 0, sin(π/3)) ; (1, 1, 1, 1)/2 V), ((cos(2π/3), 0, 0, sin(2π/3)) ; (−1, 1, 1, 1)/2 V), ((0, 1, 0, 0) ; (1, 1, 0, 0)/√2 V), ((0, cos(π/3), sin(π/3), 0) ; (1, 0, 0, 1)/√2 V), ((0, cos(2π/3), sin(2π/3), 0) ; (1, 0, 1, 0)/√2 V).
Lemmas 8 and 9 imply that the group has two isotropy types of subgroups Σ satisfying dim Fix Σ = 2. They are generated either by an order three element,
ϵ(s, r) = ((cos(π/3), 0, 0, sin(π/3)); (1, (−1)s, (−1)r, (−1)s+r)/2),
where r, s = 0, 1, (the proof that dim Fix < ϵ >= 2 follows from lemma 8) or by an order two element, a plane reflection
κ(q, n, t) = (−1)q((0, cos(nπ/3), sin(nπ/3), 0); pn((0, 0, (−1)t, 1)/√2) ),
where n = 0, 1, 2, q, t = 0, 1 and p is the permutation p(a, b, c, d) = (a, c, d, b). The isotropy planes can be labelled as follows:
P1(s, r) = Fix Σ1(s, r), where Σ1(s, r) =< ϵ(s, r) >,
P2(q, n, t) = Fix Σ2(q, n, t), where Σ2(q, n, t) =< κ(q, n, t) > .
Each of these plane contains exactly one copy of each of two types of (not conjugate) sym-metry axes, the isotropy groups of which are isomorphic to D3. They are:
L1(s, r) is the intersection of P1(s, r), P2(0, 0, s + 1), P2(s + 1, 1, r) and P2(r, 2, r + 1),
(Intersections of P1 and P2 or of two P2’s follows from lemma 7.) Since N (Σ1)/Σ1 ∼=D2, the
axes L1(s, r) and L2(s, r) intersect orthogonally.
Now, if
(i) there exist two steady states ξ1 ∈ L1 and ξ2 ∈ L2;
(ii) ξ1 is a saddle and ξ2 is a sink in P1, moreover a saddle-sink heteroclinic orbit κ1 exists
between ξ1 and ξ2 in P1;
(iii) ξ1 is a sink and ξ2 is a saddle in P2, moreover a saddle-sink heteroclinic orbit κ2 exists
between ξ1 and ξ2 in P1,
then there exists a pseudo-simple robust heteroclinic cycle ξ2 → ξ1 → ξ2. If in addition the
unstable manifold of ξ2 in P2is included in the stable manifold of ξ1: (Wu(ξ2)∩P1)⊂ Ws(ξ1),
then the system possesses a heteroclinic network comprised of two distinct (i.e. not related by any symmetry) connections ξ2 → ξ1 and one connection ξ1 → ξ2. With minor modifications
of Lemma 5 in [13], it can be proven that the conditions (i)-(iii) above are satisfied for an open set of Γ-equivariant vector fields. In the following subsection we build an explicit system with these properties.
6.3
Equivariant structure
Here we derive a third order normal form commuting with the action of the group Γ = (D3| Z2;O | V) introduced in subsection 6.2. For convenience, we write the quaternion q =
(q1, q2, q3, q4) as a pair of complex numbers,
u = (u1, u2), where u1 = q1+ iq2 and u2 = q3+ iq4. (55)
The operation of multiplication (53) takes the form
hu = (h1u1− h2u¯2, h1u2+ h2u¯1) (56)
and ˜u = (¯u1,−u2) is the conjugate of u. For quaternions presented as (55) and points in
R4 =C2 as z = (z
1, z2), the action (l; r) : z→ lzr−1 of (some) elements of Γ is
(l; r) z → lzr−1
((0, 1, 0, 0); (1, 1, 0, 0)/√2) (z1, z2)→ (eπi/4z1, e3πi/4z2)
κ(1, 0, 1) = ((0, 1, 0, 0); (0, 0, 1,−1)/√2) (z1, z2)→ (e3πi/4z2, e−3πi/4z1)
κ(0, 0, 1) = ((0, 1, 0, 0); (0, 0,−1, 1)/√2) (z1, z2)→ −(e3πi/4z2, e−3πi/4z1)
ϵ(0, 0) = ((1, 0, 0,√3)/2; (1, 1, 1, 1)/2) (z1, z2)→ (e−πi/4z1+ e−πi/4z2− i
√
3(e−πi/4z¯1+ e−πi/4z¯2;
−eπi/4z 1+ eπi/4z2+ i √ 3(eπi/4z¯ 1+ eπi/4z¯2). (57)
Using the computer algebra program GAP, we derive from (57) that the cubic equivariant terms are of the form
( ˙z1 ˙z2 ) = b(z1z¯1+ z2z¯2) ( z1 z2 ) + cΘ1+ dΘ2+ eΘ3
where b, c, d, e are real and the maps Θj are cubic expressions of z1, ¯z1, z2, ¯z2:
Θ1 = ( √ 3¯z2 1z2−2iz21z¯1+ iz1z2z¯2− 3i2z¯1z¯22 −√3z1z¯22− i 2z 2 2z¯2+ iz1z¯1z2− 3i2z¯21z¯2 ) Θ2 = ( z23− 3¯z12z2+ 2i √ 3¯z1z¯22 −z3 1 + 3z1z¯22+ 2i √ 3¯z12z¯2 ) Θ3 = ( −3√3z2 1z¯1− √ 3¯z1z¯22+ iz32 + 3i¯z12z2 −3√3z2 2z¯2− √ 3¯z2 1z¯2− iz31 − 3iz1z¯22 ) In another arrangement of the cubic monomials, the equations read:
˙z1 = (µ + b(z1z¯1+ z2z¯2))z1+ Az12z¯1+ icz1z2z¯2+ Bz23+ C ¯z12z2+ D¯z1z¯22 ˙z2 = (µ + b(z1z¯1+ z2z¯2))z2+ Az22z¯2+ icz1z¯1z2− Bz13− Cz1z¯22+ D¯z12z¯2 (58) where A =−3√3e− ic 2, B = d + ie, C =−3d + √
3c + 3ie, D = √3(−e + i(2d −
√
3
2 c)). (59) We set µ = 1 in order to make the origin unstable.
From (57) we obtain that the invariant planes are:
P2(1, 0, 1) = Fix (((0, 1, 0, 0); (0, 0, 1,−1)/ √ 2)) = (e3πi/4w, w) P2(0, 0, 1) = Fix (((0, 1, 0, 0); (0, 0,−1, 1)/ √ 2)) = (−e3πi/4w, w) P1(0, 0) = Fix (((1, 0, 0, √ 3)/2; (1, 1, 1, 1)/2)) = (√2e−3πi/4w−√3 ¯w, w)
and the invariant axes are given by
L1(0, 0) = P1(0, 0)∩ P2(0, 0, 1) = (−e3πi/4r1eα1, r1eα1), where e2α1 = −1 +
√ 2i √ 3 e πi/4, and L2(0, 0) = P1(0, 0)∩ P2(1, 0, 1) = (e3πi/4r2eα2, r2eα2), where e2α2 = 1 +√2i √ 3 e πi/4.
The system (58) restricted onto the axes L1 is
˙r1 = r1(µ + r21(2b + 2 √ 2c− 8 √ 2 √ 3 d− 4 √ 3e)),
